Correlation detection in trees for planted graph alignment

TL;DR

This paper investigates the problem of detecting correlation in trees related to graph alignment, introduces an algorithm called MPAlign, and explores the computational feasibility of graph alignment based on tree correlation detection.

Contribution

It introduces a new hypothesis testing framework for tree correlation detection, proposes MPAlign for graph alignment, and links tree detection feasibility to the complexity of graph alignment.

Findings

MPAlign succeeds in polynomial time when tree detection is feasible.

New parameter ranges identified where partial graph alignment is computationally feasible.

Conjecture that graph alignment is hard when tree detection is impossible.

Abstract

Motivated by alignment of correlated sparse random graphs, we introduce a hypothesis testing problem of deciding whether or not two random trees are correlated. We obtain sufficient conditions under which this testing is impossible or feasible. We propose MPAlign, a message-passing algorithm for graph alignment inspired by the tree correlation detection problem. We prove MPAlign to succeed in polynomial time at partial alignment whenever tree detection is feasible. As a result our analysis of tree detection reveals new ranges of parameters for which partial alignment of sparse random graphs is feasible in polynomial time. We then conjecture that graph alignment is not feasible in polynomial time when the associated tree detection problem is impossible. If true, this conjecture together with our sufficient conditions on tree detection impossibility would imply the existence of a hard phase for graph alignment, i.e. a parameter range where alignment cannot be done in polynomial time even though it is known to be feasible in non-polynomial time.